The seminar is joint for the division of Analysis and Probability and its main themes are Mathematical Physics, Probability Theory and Harmonic and Functional Analysis. The seminar is encouraged to be aimed at a broader audience, but it may be of a more specialised nature when indicated in the last line of the abstract.
The talks in the seminar are usually 60 minutes, but can also be in the format 2x45 minutes when indicated in the abstract.

The seminar takes place Tuesdays at 1315-1415 in MV:L15 unless otherwise indicated.

Should you have any questions or suggestions, feel free to email one of the organizers Martin Hallnäs (Mathematical Physics), Erik Broman (Probability Theory) or Magnus Goffeng (Harmonic and Functional Analysis).

We completely characterize the boundedness of the Volterra type integration operators acting from the weighted Bergman spaces to the Hardy spaces of the unit ball. A partial solution to this problem in the one-dimensional setting was previously obtained by Zhijian Wu. We solve the missing cases and extend the results to all dimensions. Our tools involve area methods from harmonic analysis, Carleson measures and Kahane-Khinchine type inequalities, as well as techniques and integral estimates related to Hardy and Bergman spaces. This talk is based on a joint paper with Santeri Miihkinen, Jordi Pau and Maofa Wang.

The talk is aimed at an audience

Solutions to elliptic boundary value problems on polygonal domains exhibit singularities at edges and corners. On the other hand, there has been much recent interest in nonlocal equations, e.g. involving the fractional Laplacian. We discuss a unified approach to their analysis, based on pseudodifferential techniques developed for elliptic differential equations with mixed boundary conditions. For the fractional Dirichlet problem in a domain with smooth boundary, our approach recovers recent results of G. Grubb, using independent methods. In addition to the analysis, applications of the results to robotics and the design of efficient numerical methods will be mentioned. (Joint with Rafe Mazzeo and Jakub Stocek.)
Should you have any questions or suggestions, feel free to email one of the organizers Martin Hallnäs (Mathematical Physics), Erik Broman (Probability Theory) or Magnus Goffeng (Harmonic and Functional Analysis).

**2019-10-22**

Antti Perälä (Chalmers/GU)Antti Perälä (Chalmers/GU)

*Integration operators from Bergman spaces to Hardy spaces of the unit ball*We completely characterize the boundedness of the Volterra type integration operators acting from the weighted Bergman spaces to the Hardy spaces of the unit ball. A partial solution to this problem in the one-dimensional setting was previously obtained by Zhijian Wu. We solve the missing cases and extend the results to all dimensions. Our tools involve area methods from harmonic analysis, Carleson measures and Kahane-Khinchine type inequalities, as well as techniques and integral estimates related to Hardy and Bergman spaces. This talk is based on a joint paper with Santeri Miihkinen, Jordi Pau and Maofa Wang.

The talk is aimed at an audience

**familiar with holomorphic function spaces and operator theory**. (The key concepts will be explained during the talk.)**2019-10-29**

Nikoletta Louca and Heiko Gimperlein (Heriott-Watt University, Edinburgh)Nikoletta Louca and Heiko Gimperlein (Heriott-Watt University, Edinburgh)

*Boundary problems for the fractional Laplacian: a new approach*The talk should be accessible and of interest to a general analysis audience. It will also discuss applied and numerical implications.

**2019-11-07 (Obs: Torsdag)**

Jakob Björnberg (Chalmers/GU)

Jakob Björnberg (Chalmers/GU)

*TBA*

**2019-11-12**

Bernt Wennberg (Chalmers/GU)

Bernt Wennberg (Chalmers/GU)

*TBA*

**2019-11-19**

Alexey Kuzmin (Chalmers/GU)

Alexey Kuzmin (Chalmers/GU)

*TBA*

**2019-12-03 (13.15-15.00)**

Grigori Rozenblioum (Chalmers/GU)

Grigori Rozenblioum (Chalmers/GU)

Spectral properties of the Neumann-Poincaré operator in electrostatics and elasticity

**2019-12-10**

Erik Duse (KTH)

Erik Duse (KTH)

*TBA*

**Previous seminars**

**2019-01-15**

Ramiz Reza

Ramiz Reza

*A class of Hausdorff moment sequences and its application*

We consider a class of sequences, namely q(n)/p(n), where q and p are polynomial with real coefficient and degree of q is less than degree of p. We try to investigate when such sequence is a Hausdorff moment sequence. We provide a necessary condition ( involving zeros of p ) for 1/p(n) to be a Hausdorff moment sequence. Using this necessary condition we show that Schur product of Szego kernel on unit disc with a subnormal reproducing kernel need not be a subnormal kernel. Consider the class of all real polynomial p of degree 5 such that all of its root has same real part. We provide a complete description of all polynomial p in this class so that 1/p(n) becomes a Hausdorff moment sequence.

The talk is aimed at a general analysis/probability audience.

**2019-01-22 in MV:L15**

Przemysław Ohrysko

Przemysław Ohrysko

*Spectral radius formula for Fourier-Stieltjes algebras*

In my talk I will present a short proof of the spectral radiu formula for a measure algebra on a locally compact Abelian group which can be easily generalized to Fourier-Stieltjes algebras of non-commutative locally compact groups. The second part will be devoted to proving that there are no non-trivial relations between the supremum of the Fourier-Stieltjes transform, the spectral radius and the norm of the measure. The talk is based on a joint work with Maria Roginskaya (preprint is available on arxiv.org with the identifier: 1811.11560).

**2019-01-29 in MV:L15**

Peter Sjögren

Peter Sjögren

*Sharp estimates of the spherical heat kernel*

We prove sharp two-sided global estimates for the heat kernel associated with a Euclidean sphere of arbitrary dimension. The proof relies on the classical ultraspherical polynomials and the corresponding heat kernel.
This is joint work with A. Nowak and T. Z. Szarek.
The seminar will be short, at most 45 minutes.

It is comprehensible with only very basic knowledge about the heat equation. It will be helpful but not necessary to have heard about ultraspherical (also called Gegenbauer) polynomials.

**2019-02-05**

Mateusz Wasilewski (Katholieke Universiteit Leuven)

Mateusz Wasilewski (Katholieke Universiteit Leuven)

*Quantum isomorphisms of graphs and monoidal equivalence*

I will talk about a graph isomorphism game and how different frameworks of quantum strategies lead to several notions of a quantum isomorphism between graphs; these will all be encompassed by a certain*-algebra associated to the game. Next I will introduce the quantum automorphism group of a graph and define an equivalence between these objects called the monoidal equivalence. Then I will present a connection between the *-algebra associated to the game and a *-algebra associated to monoidal equivalence, called the linking algebra, and how it allows to prove that a very weak form of quantum isomorphism automatically implies a much stronger version thereof. If time permits, I plan to discuss some open problems. This is joint work with Brannan, Chirvasitu, Eifler, Harris, Paulsen, and Su.

**2019-02-19**

Hjalmar Rosengren

Hjalmar Rosengren

*Elliptic lattice models*

In statistical mechanics, there are many solvable or integrable lattice models with intriguing mathematical properties. Typically, the most general cases of these models are described by elliptic functions. I will give an introduction to this topic, focusing on the fundamental examples of the eight-vertex model and the XYZ spin chain. Most of the talk will review classical work of Baxter and Sutherland from the 1970s. If time permits, I will briefly discuss ongoing joint work with Christian Hagendorf on nearest neighbour correlations of the supersymmetric XYZ chain.

**2019-03-05**

Stephan Wagner (Stellenbosch University)

Stephan Wagner (Stellenbosch University)

*Loop models on a fractal*

We consider two types of loop models on self-similar, fractal-like graphs, the Sierpiński gasket and its finite approximations being a classical example. In one model, a 2-factor (spanning subgraph whose components are cycles) is chosen uniformly at random. In the other, the edge set is partitioned into cycles, again uniformly at random. The presence of "holes" in the graph turns out to have interesting consequences. While the latter model mostly yields rather short cycles, long cycles surrounding the holes appear with high probability in random 2-factors, and those long loops feature interesting geometric properties reminiscent of random walks and their loop-erased variant. Moreover, an interesting phase transition can be observed.

**2019-04-02**

Yilin Wang (ETH Zürich)

Yilin Wang (ETH Zürich)

*Loewner energy, determinants of Laplacians and Brownian loop measure*

We have introduced Loewner energy to describe the large deviation behaviour of chordal Schramm-Loewner evolution with vanishing parameter. The parametrization-independence of its generalization to Jordan curves was explained by an intrinsic description using zeta-regularized determinants of Laplacians which can be interpreted as the mass of Brownian loop measure. This identity suggests that one may interpret the Loewner energy as the total mass of the Brownian loops attached to the curve. I will show that the cut-off by equipotentials provides an appropriate renormalization to make sense of this statement.

**2019-04-09**

Lachlan MacDonald (University of Wollongong)

Lachlan MacDonald (University of Wollongong)

*Dynamical invariants of foliated manifolds*

A foliation of a manifold M specifies the set of integral submanifolds (referred to in foliation theory as "leaves") to some nonsingular system of differential equations on M. The "holonomy" of integral curves in a foliated manifold gives rise to an intrinsic dynamical system, whose topology has implications for the global behaviour of solutions to the corresponding system of DEs. In this talk, I will give a primer on foliations and their associated dynamical invariants, as well as discuss how this information can be accessed in the spirit of the Atiyah-Singer index theorem.

The talk is aimed at an audience familiar with the basics of manifolds and vector bundles. Some knowledge of algebraic topology and operator algebras will be helpful but not necessary.

**2019-05-07**

Michael Björklund

Michael Björklund

*Fun with i.ni.d. random variables*

Given a countable group G and a function f : G \ra (0,1) we consider a sequence (x_g) of independent {0,1} random variables, indexed by g \in G, where P(x_g = 0) = f(g). If f is constant, or “close to” a constant, then this sequence is “essentially” stationary. If f is varying very rapidly, then no “stationary” behaviour can be seen in the sequence. However, on a large class of groups (namely those with non-trivial first L^2-cohomology), there are interesting choices of f, which are far from constant, but for which some “stationary” behaviour is retained. We shall discuss this class of groups and the various f’s that give rise to interesting “almost stationary”, but not stationary, processes.

**2019-05-16 (Thursday) at 10:00 in MV:L15**

Andreas Rosén

Andreas Rosén

*Maxwell scattering with the help of Dirac*

This talk concerns numerical computation of the electromagnetic field scattered by in general non-smooth objects, through the methods of integral equations on the boundary. There are three main steps: 1. Integral formulation. 2. Discretization 3. Fast solvability of dense linear systems. We are primarily concerned with 1, and report on work in progress joint with Johan Helsing (Lund), using his state-of-the-art algorithms for 2. The new integral formulations

we are testing are obtained using Clifford algebra and hypercomplex Hardy spaces. In this first test we have considered two-dimensional TM scattering, and in particular the computation of surface plasmon excitations by corner singularities. For the integral equation this means that we hit the essential spectrum.

**2019-05-20 (Monday) at 13:15**

Nikoletta Louca (Heriot-Watt University, Edinburgh)

Nikoletta Louca (Heriot-Watt University, Edinburgh)

*The magnitude function on domains*

The notion of magnitude is a real-valued invariant of metric spaces that has been introduced by Leinster, and is analogous to the Euler characteristic of topological spaces and the cardinality of sets. Basic questions about the geometric properties of the magnitude function still remain open, even for compact subsets of R^n. In this talk we discuss some of the properties of the magnitude function on smooth domains of (mainly odd-dimensional) manifolds, part of which is work in progress with Heiko Gimperlein and Magnus Goffeng.

**2019-05-21**

Przemysław Ohrysko

Przemysław Ohrysko

*Spectrally reasonable measures II*

Let G be a locally compact Abelian group. A complex measure on G is said to have a natural spectrum if its spectrum (in a standard Banach algebras sense) is equal to the closure of the range of its Fourier-Stieltjes transform. It is well-known that on every locally compact non-discrete Abelian group there exists a measure with non-natural spectrum (Wiener-Pitt phenomenon). The structure of the set of all measures with a natural spectrum was studied by Hatori and Sato who proved that this set span the whole algebra of measures and hence cannot be closed under addition. Inspired by this result, the notion of spectrally reasonable measure was introduced as a measure which perturbs any measure with a natural spectrum to a measure with a natural spectrum. It turned out that the set of all spectrally reasonable measures has Banach algebra structure.

In my talk, I will present the recent developments in this area: the complete characterization of spectrally reasonable measures for classical groups of harmonic analysis – the Euclidean spaces and the torus. The first of the presentation will be devoted to the general discussion and advertisement of the results and in the second part I will show the details of the proof of the classification of spectrally reasonable measures for Euclidean spaces. The arguments are quite involved and use the tools from classical harmonic analysis, Banach algebra theory and some nearly completely forgotten results from the general topology.

The talk is based on the preprint with the same title, available on arxiv.org with the identifier: 1903.04853 written in collaboration with Michał Wojciechowski.

**2019-05-28**

Ksenia Fedosova (University of Freiburg)

Ksenia Fedosova (University of Freiburg)

*Eisenstein series and Selberg zeta function twisted by representations with non-expanding cusp monodromy*

In this talk, we investigate the behaviour of the Eisenstein series, or generalized eigenfunctions of the Laplace operator on hyperbolicsurfaces. We twist them by a (possibly) non-unitary representation of the fundamental group of the manifold, show their convergence on some half-plane and study their Fourier expansion. Further, we show the convergence and meromorphic continuation of the Selberg zeta function twisted by the same type of representations.

**2019-06-05 (Wednesday) at 10:00**

Jörg Weber (University of Bayreuth)

Jörg Weber (University of Bayreuth)

*The Relativistic Vlasov-Maxwell System with Boundary Conditions and External Currents*

The time evolution of a collisionless plasma is modelled by the relativistic Vlasov-Maxwell system, which couples the Vlasov equation (the transport equation) with the Maxwell equations of electrodynamics. We consider the case that the plasma particles are located in a bounded domain, for example a fusion reactor. In the exterior, there are external currents that may serve as a control of the plasma if adjusted suitably. Also, we allow material parameters, namely permittivity and permeability, which may depend on the space coordinate.

In the first part of the talk, we discuss the solution theory of the modeling nonlinear PDE system and prove existence of global-in-time weak solutions.

In the second part, we consider a minimizing problem: A typical aim is to drive the amount of particles hitting the boundary of their container and thus causing damage to a minimum, while the control costs should be kept as small as possible. We prove existence of minimisers of the arising minimizing problem and give an approach to derive first order optimality conditions.

**2019-06-11**

Roger Behrend (Cardiff University)

Roger Behrend (Cardiff University)

*The combinatorics of alternating sign matrices*

An alternating sign matrix (ASM) is a square matrix in which each entry is -1, 0 or 1, and along each row and column the nonzero entries alternate in sign, starting and ending with a 1. It was conjectured by Mills, Robbins and Rumsey in 1982 that the number of ASMs of fixed size is given by a certain simple product formula. A relatively short proof of this conjecture was obtained by Kuperberg in 1996, using connections with the six-vertex model of statistical mechanics. It was also conjectured in the 1980s that the number of ASMs of fixed odd size which are invariant under diagonal and antidiagonal reflection is given by a simple product formula. In this talk, I will provide an introduction to ASMs, a review of Kuperberg's proof of the ASM enumeration formula, and an outline of a recent proof of the enumeration formula for diagonally and antidiagonally symmetric ASMs of odd size.

The talk is aimed at the whole division.

**2019-08-27**

Antar Bandyopadhyay (Indian Statistical Institute (Delhi Centre))

Antar Bandyopadhyay (Indian Statistical Institute (Delhi Centre))

*Random Recursive Tree, Branching Markov Chains and Urn Models*

In this talk, we will show that a branching Markov chain defined on the random recursive tree is nothing but a balanced urn model. This is a novel connection between these two apparently unrelated probabilistic models. Exploring the connection further we will derive fairly general scaling limits for balanced urn models with colors indexed by any Polish Space. Thus generalizing the classical urn models defined on finitely many colors to a measure valued process on a general Polish Space. We will use the connection to show that several exiting results on classical finite color urn schemes, as well as, recently introduced infinite color urn schemes, can be derived easily from the general asymptotic. In particular, we will prove that the well know classical result of the finitely many color balanced urn, when the replacement matrix is irreducible will also holds for countably infinite many colors (under certain technical assumptions). We will also discuss the special case, of null recurrent replacement matrices, which can only arise in the context of infinitely many colors. We will further show that the connection can also be used to derive exact asymptotic of the sizes of the connected components of a random recursive forest, which is obtained by removing the root of a random recursive tree.

(Parts are joint work with Debleena Thacker and Svante Janson)

**2019-09-10**

Joakim Arnlind (Linköpings Universitet)

Joakim Arnlind (Linköpings Universitet)

*Pseudo-Riemannian calculi and noncommutative connections*

In a series of papers, we have developed Pseudo-Riemannian calculi as a framework for discussing Levi-Civita connections and curvature in noncommutative geometry. One of the original motivations was to understand under what conditions a torsion-free and metric connection is unique, and to what extent the curvature tensor exhibits the classical symmetries. Moreover, it turns out that one may develop a theory of submanfolds and minimal embeddings in the spirit of differential geometry, finding noncommutative analogues of the second fundamental form, Gauss' equations, etc. In this talk, I will give an overview of Psuedo-Riemannian calculi together with a few concrete examples. This is joint work with Mitsuru Wilson and Axel Tiger Norkvist.

**2019-09-17**

Yanqi Qiu (Chinese Academy of Sciences)

Yanqi Qiu (Chinese Academy of Sciences)

*A class of positive functions on the circle and branching-type stationary stochastic processes*

We propose a definition of branching-type stationary stochastic processes on rooted trees. The spectral measure of such stochastic processes are related to positive definite functions on positive integers. After giving a complete criterion of such positive definite functions in the case of rooted homogeneous trees, we are going to talk about their applications on inequalities of Hankel operators. The talk is based on joint work with Zipeng Wang.

This talk is aimed for probability and analysis audience.

**2019-09-24 at 10am in MV:L14**

Joint AP/KASS Seminar:

Richard Schoen (University of California, Irvine)

Joint AP/KASS Seminar:

Richard Schoen (University of California, Irvine)

*Positive scalar curvature, the Dirac operator, and minimal hypersurfaces*

We will discuss the approaches which have been successful in understanding manifolds with positive scalar curvature. Two basic and related questions one can ask are the topological classification of such manifolds and the positive mass theorem. The two approaches which have worked in all dimensions are the Dirac operator and the minimal hypersurface approach. Both have many successes but also limitations which we will describe. In the case of three dimensions there are also the inverse mean curvature flow and the Ricci flow which yield strong results.

**2019-09-26 Thursday at 15:15 in MV:L15**

Joint AP/AGNT Seminar:

Wee Teck Gan (National University of Singapore)

Joint AP/AGNT Seminar:

Wee Teck Gan (National University of Singapore)

*Triality and functoriality*

I will discuss some implications of the existence of the triality automorphism in the Langlands program, such as the demonstration of certain instances of functorial lifting. This is joint work with Gaetan Chenevier.

**2019-10-08 Tuesday at 1515**

Andreas Juhl (Humboldt-Universität Berlin)

Andreas Juhl (Humboldt-Universität Berlin)

*Branson-Gover operators and residue families.*

In 2004, T. Branson and R. Gover introduced differential operators acting on differential forms which generalize the conformal powers of the Laplacian (GJMS operators). Their construction rests on tractor calculus. We found that these constructions admit an alternative description in terms of a generalization of the notion of residue families. In the scalar case, this notion has been introduced by the author in 2009. In the form case, residue families are conformally covariant one-parameter families of differential operators which are curved versions of so-called symmetry breaking operators (a concept recently introduced by T. Kobayashi). In the seminar, we describe these concepts, their interactions and some of the emerging new results.

**2019-10-15**

Joao Pedro Paulos (Chalmers/GU)

Joao Pedro Paulos (Chalmers/GU)

*Classical Descriptive Set Theory and Closed Sets of Uniqueness*

Descriptive Set Theory (DST) is the study of Polish spaces and its subsets. Even though its ethos is indissociable from questions coming from Set Theory and Logic, the development of DST lead to many applications within a wide range of topics emerging from Harmonic Analysis, Recursion Theory, Functional Analysis or Dynamical Systems. In this talk, we introduce some elementary concepts and prove a result on the complexity of the set of closed sets of uniqueness

The talk is aimed at a general analysis/probability audience.